The non-Hermiticity of \(H\) 
in the working space \({\cal H}_{math}\) 
leads to
the necessity of using some
less straightforward methods of solution
of the underlying Schrödinger equation.
In many models,
the merits of the adaptability of the
non-unitary
mapping
(often called, in the light of papers [1]},
Dyson map) proved to
prevail
(cf., e.g., the success and productivity of this approach in
nuclear physics [2]}).
Still, one of the unavoidable consequences
of the non-unitarity of \(\Omega \) 
is that
the conventional interpretation of the
eigenstates of \(H\) 
has to be
modified.
In the manner described in reviews [3]}, [4]}
the situation may be clarified and
visualized using the following diagram
\(\begin{array}{c}\ \ \begin{array}{|c|}\hline \ {\rm inaccessible \ physical\ Hilbert\ space} \ {\cal L}\ {\rm of\ textbooks} \\\hline \end{array}\\ \ \ \ \ \ \ \ {\rm (Dyson^{\prime }s\ map}\ \Omega ) \ \nearrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nwarrow \searrow \ { \rm (equivalence)}\ \ \ \\\ \ \begin{array}{|c|}\hline \ {\rm friendlier\ representation \ space} \ {\cal H}_{math}\ \\\hline \end{array}\ \stackrel{{\rm (metric}\ \Theta )}{ \longrightarrow }\ \begin{array}{|c|}\hline \ {\rm alternative \ physical\ space} \ {\cal H}_{phys}\ \\\hline \end{array}\\\\\end{array}\) 
